\newproblem{lay:2_3_41}{
  % Problem identification
	\begin{large}
    \textbf{Lay, 2.3.41}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
  Suppose an experiment leads to the following system of equations
	\begin{center}
		$4.5x_1+3.1x_2=19.249$\\
		$1.6x_1+1.1x_2=6.843$
	\end{center}
	\begin{enumerate}[a.]
		\item Solve the previous equation system, and then, the equation system below in which the data on the right has been rounded to two decimal places.\\
			\begin{center}
				$4.5x_1+3.1x_2=19.25$\\
				$1.6x_1+1.1x_2=6.84$
			\end{center}
		\item The entries in the rounded system of equations differ from those of the exact system by less than 0.05\%. Find the percentage error when using the
		   solution of the rounded equation system as an approximation to the solution of the exact system.
	\end{enumerate}
}{
  % Solution
	\begin{enumerate}[a.]
		\item The solution of the exact equation system is \\
				  \begin{center}$\mathbf{x}_{\mathrm{exact}}=A^{-1}\begin{pmatrix}19.249\\6.843\end{pmatrix}=\begin{pmatrix}3.94\\0.49\end{pmatrix}$\end{center}.
		      The solution of the rounded equation system is \\
					\begin{center}$\mathbf{x}_{\mathrm{rounded}}=A^{-1}\begin{pmatrix}19.25\\6.84\end{pmatrix}=\begin{pmatrix}2.90\\2.00\end{pmatrix}$\end{center}
		\item The error percentage is given for each variable as
		      \begin{center}
						$\epsilon_1=100\frac{|x_{1,\mathrm{exact}}-x_{1,\mathrm{rounded}}|}{|x_{1,\mathrm{exact}}|}=100\frac{|3.94-2.90|}{|3.94|}=26.40$\% \\
						$\epsilon_2=100\frac{|x_{2,\mathrm{exact}}-x_{2,\mathrm{rounded}}|}{|x_{2,\mathrm{exact}}|}=100\frac{|0.49-2.00|}{|0.49|}=308.16$\% \\
					\end{center}
	\end{enumerate}
}
\useproblem{lay:2_3_41}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
